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\begin{document}

\title{DSR Simulation Review}%
\author{Xuezhi Liu\\
        November 18, 2011} %\footnote{Financial Stability Department, email: xliu@bankofcanada.ca}
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%\email{xliu@bankofcanada.ca}%

\thanks{Xuezhi Liu, Financial Stability Department, email: xliu@bankofcanada.ca}%
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%\begin{abstract}

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\section*{DSR Framework ~\cite{Framework}}

%DSR Framework

\begin{enumerate}

\item
    DSR distribution:
    \begin{align}%\label{}
        \text{DSR distribution} &= F\left(\text{Income, Debt, Interest rates,} \right. \nonumber\\
        &\qquad \left. \text{Other HH factors} \right).
    \end{align}
\\

\item
    To calculate the micro DSR,
    \begin{align}
        DSR &= \frac{\sum \text{Payments}}{\text{Gross Income}} \nonumber\\
            &= \frac{\sum ( \text{Principal Repayment} + \text{Interest Repayment})}{\text{Gross Income}}.
    \end{align}
\\

\item
    The amount of principal repayment due is:
    \begin{equation} %\label{}
        \text{Share} \_ \text{Principal} = PC - \text{Interest} = PC - SC \times ir ,
    \end{equation}
    where \( PC \) represents a HH's total annual loan payments, \( SC \) is its current credit balance and \(ir\) is the applicable interest rate.
\\

\item
    Principal repayments are set as a constant share of credit balance
    \begin{equation}
        \text{Share} \_ \text{Principal} = \nicefrac{\text{Principal}}{SC}.
    \end{equation}
\\


\item
    Therefore, the annual loan payment is,
    \begin{align}
        PC &= \text{Principal} + \text{Interest} \nonumber \\
        &= SC \times \text{Share} \_ \text{Principal} + SC \times ir \nonumber \\
        &= SC \times (\text{Share} \_ \text{Principal} + ir).
    \end{align}
\\

\item
    Income growth dynamics:
    \begin{equation}
        \text{Income} \sim \mathcal{N}(r_j,\sigma_j),
    \end{equation}
    where \\
    \indent $j = \text{HH income class, which is } 1,2,3,4,5$; \\
    \indent $r_j = $ average income growth of HH in class $j$; \\
    \indent $\sigma_j = $ estimated standard deviation of income growth for HH in class $j$.
\\

\item
    Growth of total HH debt:
    \begin{align} \label{eq:eqn7}
        \Delta TC_t &= c_{11} + \alpha_{11} \Delta r_t + \alpha_{12} \Delta i_t + \alpha_{31} (1 + hp_t) HW_{t-1} \mathbb{I}_0 + \nonumber\\
          & \lambda_1 (c_{11} + \alpha_{11} \Delta r_t + \alpha_{12} \Delta i_t + \alpha_{31} (1 + hp_t) HW_{t-1}) D40 + \epsilon_1,
    \end{align}
    Growth of mortgage debt:
    \begin{align} \label{eq:eqn8}
        \Delta MC_t &= c_{12} + \alpha_{12} \Delta r_t + \alpha_{22} \Delta i_t + \alpha_{32} (1 + hp_t) HW_{t-1} + \nonumber\\
          & \lambda_1 (c_{12} + \alpha_{12} \Delta r_t + \alpha_{22} \Delta i_t + \alpha_{32} (1 + hp_t) HW_{t-1}) D40 + \epsilon_2,
    \end{align}
    where\\
    \indent $\Delta$: first-difference operator; \\
    \indent $i$: interest rate; \\
    \indent $r = \ln(\text{HH Income})$; \\
    \indent $hp$: house values; \\
    \indent $\mathbb{I}_0$: 1 if HH has a mortgage, 0 otherwise; \\
    \indent $HW = \ln(\text{housing wealth})$, where housing wealth is the difference between the value of the house and the amount of the mortgage; \\
    \indent $D40$: dummy variable indicating whether the household has a DSR variable above 40\% threshold.
\\

\item
    Growth of consumer or mortgage debt:
    \begin{equation}
	\Delta C_t = \frac{\sum (1 + \Delta C_{it}) w_i C_{it-1} - \sum w_i C_{it-1}}{\sum w_i C_{it-1}}.
    \end{equation}
    Adjusted individual growth of consumer credit and mortgage consistent with equations \ref{eq:eqn7} and \ref{eq:eqn8} and the aggregate scenario:
    \begin{equation}
	\Delta C_{\text{1} it} = (AG - \Delta C_t) + \Delta C_{it},
    \end{equation}
    where \\
    \indent $i$: household; \\
    \indent $\Delta C_{it}$: individual growth on consumer and mortgage debt implied by equations \ref{eq:eqn7} and \ref{eq:eqn8}; \\
    \indent $AG$: average aggregate growth assumed adjusted for the first-time homebuyers.
\\

\item
    Assume the duration of unemployment follows a chi-squared distribution:
    \begin{equation}
        \text{UnempDure} \sim \chi^2_k.
    \end{equation}
\\

\item
    Unemployment rate:
    \begin{equation}
        U_t = \frac{\sum_i \text{\# of weeks unemployed per household}_i}{\sum_i \text{Total \# of weeks in the labour force}_i},
    \end{equation}
where, $i$: individuals in the labour force over year period.
\\

\item
    The probability of default (PD):
    \begin{equation}
        PD_t = \frac{\sum_i \text{HH that defaulted on any loan for more than three months}_i}{\text{Total \# of HHs}_i}.
    \end{equation}
\\

\item
    Mortgage rate with maturity $y$ at period $t$:
    \begin{equation}
	   i_{y,t} = \text{ovn}_t + \text{risk premium}_{y,t} + \text{term premium}_{y,t},
    \end{equation}
    where \\
    \indent ovn$_t$: overnight rate or policy rate;\\
    \indent risk premium$_{y,t}$: aggregate risk premium;\\
    \indent term premium$_{y,t}$: aggregate term premium.
\\

\item
    Condition for default:
    \begin{align}
       \mathbb{I}_{\text{def}}
            &= \mathbb{I}_{\left\{ \frac{\text{LiqAss} - \text{UnempDure} \times (\text{DbtPmt} - \frac{1}{2} \frac{\text{Income}}{52})^{+}}{\text{DbtPmt}} < -13 \right\}} \nonumber\\
            &= \mathbb{I}_{\left\{  \frac{\text{LiqAss}}{\text{DbtPmt}} - \text{UnempDure} \times \left( 1 - \frac{1}{2} \frac{1}{\text{DSR}} \frac{1}{52} \right)^{+}  < -13 \right\} } \nonumber\\
            &= \mathbb{I}_{\text{def}}\left(\cdot,\text{DSR}\right),
    \end{align}
    where \\
    \indent $()^{+} = \max(\cdot,0)$;\\
    \indent DbtPmt: weekly debt payment. \\
    \indent LiqAss: HH's total financial liquid assets, including checking \& saving account, term deposits (term + GICs), bonds (T-bills, bonds, and other guaranteed investment), stocks, and mutual funds;\\
    \\
    To be noticed, $\mathbb{I}_{\text{def}}$ is an increasing function of DSR in the domain of $\left(\frac{1}{2 \times 52},\infty\right)$.
\\

\item
    Loans-in-Arrears rate:
    \begin{align}
        \text{Loans-in-Arrears Rate}
            &= \frac{\text{TotDefDbt}}{\text{TotDbt}} = \frac{\text{TotDbt} \times \mathbb{I}_{\text{def}}}{\text{TotDbt}} \nonumber \\
            &= f(\cdot,\mathbb{I}_{\text{def}}(\cdot,\text{DSR})) \nonumber\\
            &= f(\cdot,\text{DSR}),
    \end{align}
    where \\
    \indent TotDefDbt: total defaulted debts, \\
    \indent TotDbt: total debts. \\
    \\
    Also, this is an increasing function of DSR.
\\

\end{enumerate}



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